An Exact 56-Addition, Rank-23 Scheme for General 3*3 Matrix Multiplication

We present a rank-$23$ algorithm for general $3\times 3$ matrix multiplication that uses $56$ additions/subtractions and $23$ multiplications, for a total of $79$ scalar operations in the standard bilinear straight-line model. This improves the recent sequence of $60$-, $59$-, and $58$-addition rank-$23$ schemes. The algorithm works over arbitrary associative, possibly noncommutative, coefficient rings. Its tensor coefficients are ternary, meaning that every coefficient lies in $\{-1,0,1\}$. Correctness is certified by the $729$ Brent equations over $\mathbb{Z}$, and the verifier also expands the straight-line program and performs additional finite-field and noncommutative implementation tests.